The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety K, the lattice Lq(K) possesses the Day-Pudlak-Tuma property. But if a quasivariety is locally finite (without the condition of being finitely generated), that lattice satisfies only the Pudlak-Tu̇ma property.
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